Hi,
In general, I think the methods used by Mathematica are vastly different
from those used in hand calculations. For example, a lot of integration
is done using very complicated objects known as HypergeometricPFQ and
MeierG functions. These are translated back to something simpler (where
possible) before the result is presented.
The problem with your idea is that unless you actually do a fair amount
of math by hand, you don't get the feel for what it really means and how
things work. It may seem a waste of time, but I would encourage you to
bear with it! Perhaps the lecturers should discuss this issue a bit.
Mathematica can be a great help in learning math provided that you don't
just use it to cheat. You can use it to generate examples of similar
problems to the one you are looking at, or to explore more difficult
problems.
David Bailey
http://www.dbaileyconsultancy.co.uk